
where \(\overline{\cdot }\)  represents the average over unitary gates, and \(m\)  labels the measurement results.
Using the framework developed in Ref. [1]}, [2]}, we can express \(I_{A:B|M}\)  using the replica method as the \(n \rightarrow 1\)  limit of the replicated quantities
\(I^{(n)}_{A:B|M} = \frac{2}{1-n}\log \left( \frac{\overline{\sum _m \operatorname{tr}\tilde{\rho }_{A,m}^n}}{\overline{\sum _m \operatorname{tr}\tilde{\rho }_{m}^n}}\right),\) 
